# Useless rules

In theoretical computer science, in particular in the theory of formal languages, **useless rules** of a formal grammar are those rules of symbol production that are unreachable or unproductive, that is, that can or need never be applied. Simply, they "can be removed from the grammar without affecting the language produced"[1]

## Definition

Given a context-free grammar, a nonterminal symbol *X* is called **productive**, or **generating**, if there is a derivation *X* ⇒^{*} *w* for some string *w* of terminal symbols. A nonterminal symbol *X* is called **reachable** if there is a derivation *S* ⇒^{*} α*X*β for some strings α, β of non-terminal and terminal symbols, and where *S* denotes the grammar's start symbol.

A rule with an unproductive or unreachable symbol on its left-hand side can be deleted from the grammar without changing the accepted (a.k.a. generated) language.
Likewise, an alternative containing such a symbol can be deleted from the right-hand side of a rule without changing the language.
Such rules and alternatives are called **useless**.[2]

For formal grammars that are not context-free, similar definitions apply.

## Examples

Denoting nonterminal and terminal symbols by upper- and lower-case letters, respectively,
in the following regular grammar with start symbol *S*

S → Bb | Cc | Ee |

B → Bb | b |

C → Cc | c |

D → Bd | Cd | d |

E → Ee |

the nonterminal *D* is unreachable, and *E* is unproductive.
Hence, omitting the last two rules doesn't change the language accepted by the grammar, nor does omitting the alternative "| *Ee*" from the right-hand side of the rule for *S*.

## Cleaning Useless Rules

Hopcroft, et al.[3] give an algorithm to eliminate useless rules from a context-free grammar.

Aiken and Murphy[4] give a fixpoint algorithm to detect which nonterminals of a given regular tree grammar are unproductive.

## References

- (PDF) https://www.springer.com/cda/content/document/cda_downloaddocument/9780387202488-c1.pdf?SGWID=0-0-45-466216-p52091986. Missing or empty
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(help) - John E. Hopcroft; Rajeev Motwani; Jeffrey D. Ullman (2003).
*Introduction to Automata Theory, Languages, and Computation*. Addison Wesley.; here: Sect.7.1.1, p.256 - Theorem 7.2, Sect.7.1, p.255ff
- Aiken, A.; Murphy, B. (1991). "Implementing Regular Tree Expressions".
*ACM Conference on Functional Programming Languages and Computer Architecture*. pp. 427–447. CiteSeerX 10.1.1.39.3766.; here: Sect.4